Scalar problems in junctions of rods and a plate.
II. Self-adjoint extensions and simulation models.
Abstract
In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these ”parasitic” eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.
Keywords: junction of thin rods and plate, scalar spectral problem, asymptotics, dimension reduction, self-adjoint extensions of differential operators, function space with detached asymptotics
MSC: 35B40, 35C20, 74K30.
1 Introduction
1.1 Motivations
As it was observed in [4], an asymptotic expansion of a solution of the Poisson scalar mixed boundary-value problem in a junction of thin rods and a thin plate in a certain range of physical parameters gains the rational dependence on the big parameter where is a small parameter characterizing the diameters of the rods and the thickness of the plate. Similar asymptotic forms had been discovered for other elliptic problems stated in domains with singular perturbations of the boundaries, see the books [16, 11, 9]. The aim of this paper is to study the scalar spectral problem associated to the Poisson problem studied in [4].
Based on the asymptotic procedure in [16, §2.2.4, §5.5.2, §9.1.3] and [17, 4], it is quite predictable that the asymptotic expansions of the eigenpairs ”eigenvalue/eigenfunction” become much more complicated than the one obtained for the solution of the Poisson problem and purchase the holomorphic dependence on the parameter . Such sophistication of the asymptotic expansions and the lack of algorithms allowing to clarify the appearing holomorphic functions make them almost useless in applications, especially for combined numerical and asymptotic methods.
In [14] J.-L. Lions announced as an open question an application of the technique of self-adjoint extensions of differential operators for modeling boundary-value problems with singular perturbations. This technique has been employed in [18, 19, 20] and others to deal with particular types of perturbations but at our knowledge, was never used before for the study of spectral problems for junctions of thin domains with different limit dimension like 1d and 2d for the rods and plate in our 3d junction.
The use of such techniques in the asymptotic analysis here is the main novelty of our work. We observe that to the spacial junction represented in fig. 1,a, corresponds the hybrid domain depicted in fig. 2,a, which consists of several line segments joined to some interior points of a planar domain. Supplying these elements of with differential structures, we describe a family of all self-adjoint operators which is parametrized by a finite set of free parameters and choose an appropriate set by examining the boundary layer phenomenon in the vicinity of the junction zones, namely where the rods are inserted into small sockets in the plate, fig. 1,b. As a result, we obtain a model which provides the satisfactory proximity instead of the uncomfortable one within a simplified but comprehensible version of the conventional asymptotic procedure. It should be mentioned that our model involves only one scalar integral characteristic of each junction zone.
1.2 Formulation of the spectral problem
Given a small parameter , we introduce the thin plate and rods
| (1.1) | ||||
| (1.2) |
Here, are domains in the plane with smooth (for simplicity) boundaries and the compact closures are positive numbers independent of , and , for . Reducing a characteristic size of to , we make Cartesian coordinates and all geometric parameters dimensionless. We fix some such that, for and , where is the cross section of the rod while and are its lower and upper ends. In the sequel the bound can be diminished but always remains strictly positive.
The rods (1.2) are plugged into sockets, i.e., holes in the plate, fig. 1,b and a,,
| (1.3) |
and compose the junction
| (1.4) |
The lateral side of the plate (1.1) and (1.3) and their bases are denoted by and
| (1.5) |
The lateral side of the rod is divided into two parts
where the latter is the junctional boundary of the socket.
In the junction (1.4) we consider a spectral problem consisting of the differential equations with the Laplace operator
| (1.6) | ||||
| (1.7) |
the Neumann and Dirichlet boundary conditions
| (1.8) | ||||
| (1.9) | ||||
| (1.10) |
and the transmission conditions
| (1.11) | ||||
| (1.12) |
Here, is a spectral parameter, and are restrictions of the function on and , respectively, and is the outward normal derivative at the surface of in (1.8) and (1.9) while in the second transmission condition is outward with respect to the rods. In what follows we mainly deal with the coefficients
| (1.13) |
in the most informative but complicated case
| (1.14) |
We emphasize that in the case (1.14) the limit passage in the problem (1.6)-(1.12) leads to the eigenvalue problem for a self-adjoint operator in the skeleton of (1.4) in fig. 2a, a hybrid domain, while, for and this limit operator decouples, cf fig. 2b. We discuss the latter cases in Section 5 and pay attention to the homogeneous junction with
| (1.15) |
Other cases will be discussed in Section 5. The factors and in (1.13) are real positive numbers.
The variational formulation of the problem (1.6)-(1.12) reads:
| (1.16) |
where is the Sobolev space of functions satisfying the Dirichlet conditions (1.10) on
| (1.17) | ||||
is the natural scalar product in the Lebesgue space and everywhere stands for summation over .
In view of the compact embedding the spectral problem (1.16) possesses, for every fixed , the following positive monotone unbounded sequence of eigenvalues
| (1.18) |
listed according to their multiplicity. The corresponding eigenfunctions can be subject to the normalization and orthogonality conditions
| (1.19) |
where is the Kronecker symbol.
1.3 The hybrid domain
The asymptotic analysis which has been presented at length, for example, in [4, 5, 6] and, in particular, includes the dimension reduction procedure, converts the differential equations (1.6) and (1.7) with the Neumann boundary conditions (1.8) and (1.9), respectively, into the limit equations
| (1.20) | ||||
| (1.21) |
where is the punctured domain and is the area of the domain . Moreover, a primary examination of the boundary layer phenomenon near the lateral side of the plate and the end of the rod , respectively, gives the following boundary conditions
| (1.22) | ||||
| (1.23) |
However, the one-dimensional and two-dimensional problems are not completed yet due to the lack of boundary conditions at the endpoints of the intervals and because the differential equation (1.20) is fulfilled for sure only outside the points , since near the sockets the geometrical structure of the junction (1.4) changes and becomes crucially spacial so that the dimension reduction does not work. As was shown in [4], this observation requires to consider solutions of the problem (1.20), (1.22) with logarithmic singularities at the points in the set . We also will take in the sequel such singular solutions into account, however further considerations in this paper diverge from the asymptotic analysis used in [4]. Indeed, we will provide two abstract but applicable formulations of a spectral problem in the hybrid domain in fig. 2,a, which give an approximation of the spectrum (1.18) with relatively high precision. First, we detect a self-adjoint operator as an extension of the differential operator of the problem (1.20)-(1.23) supplied (cf. (2.3) and (2.1)) with the restrictive conditions
| (1.24) |
Second, we construct certain point conditions at which tie the independent problems (1.20), (1.22) and (1.21), (1.23) into a formally self-adjoint problem in the hybrid domain These two formulations happen to be equivalent and both are realized as operators with the discrete spectrum which, in the low-frequency range, approximate the spectrum of the problem (1.6)-(1.12) (or, equivalently, (1.16)) with admissible precision111The error estimates are derived in the paper with quite simple tools. Advanced estimation may detect the accuracy and extend the proximity property of the models to a part of the mid-frequency range, cf. [20]. The latter, however, enlarges enormously massif of calculations. . Unfortunately, serving for a particular range of the spectrum, both the operators lose the positivity property and gain so called parasite eigenvalues which are negative and big, of order ; therefore, we prove that they lay outside the scope of the asymptotic models and have no relation to the original problem. In order to furnish, for example, an application of the minimum principle, cf. [2, Thm. 10.2.1], we construct detailed asymptotics of these parasite eigenvalues and of the corresponding eigenfunctions, which are located in the very vicinity of the points and decay exponentially at a distance from them.
Parameters of the self-adjoint extension and ingredients of the point conditions are found out with the help of the method of matched asymptotic expansions on the basis of special solutions described in the first part [4] of our work. Both linearly depend on and this makes the eigenvalues and eigenvectors to be real analytic functions in However, a possible numerical realization of the models with a small but fixed parameter does not require to take into account such complication of asymptotic expansions. We, of course, compute explicitly couple of initial terms of the convergent series in
1.4 Outline of the paper
In Section 2 we describe all self-adjoint extensions of the operator of the problem (1.20)-(1.24) as well as the point conditions which involve the problems (1.20), (1.22) and (1.21), (1.23) into a symmetric generalized Green formula. This material is known and is presented in a condensed form, mainly in order to introduce the notation and explain some technicalities used throughout the paper. We refer to the review papers [29], [3] and [21] for a detailed information. If the skeleton decouples in the limit, see fig. 2,b, then the extended operator of the Neumann problem (1.20), (1.22) may require for ”potentials of zero radii” [1, 29] or ”pseudo-Laplacian” in the terminology [7], but in this case the ordinary differential equations (1.21), (1.23) are supplied with either Neumann, or Dirichlet condition at the endpoints of the interval (cf., Remark 5, 6 and see the paper [3] which provides the complete description of the techniques of self-adjoint extensions).
The most interesting situation occurs under the restriction (1.14) when the skeleton does not decouple in the limit, see fig. 2,a. The asymptotic procedure in [4] allow us to determine in Section 3 appropriate parameters of a particular self-adjoint extension serving for the original problem (1.6)-(1.12) as well as all ingredients of the point condition in the corresponding hybrid model.
The most cumbersome part of our analysis is concentrated in Section 4, where we perform the justification of our asymptotic models with the help of weighted estimates obtained in [4]. We state here the main result of the paper, Theorem 16. Section 5 contains some simple asymptotic formulas and the example of the homogeneous junction, see (1.15).
2 General statement of problems on the hybrid domain
2.1 Unbounded operators and their adjoints
Let be an unbounded operator in the Lebesgue space with the differential expression and the domain
| (2.1) |
The operator is symmetric and closed. By a direct calculation, it follows that the adjoint operator gets the same differential expression but its domain is bigger, namely
| (2.2) |
Hence, and the defect index of is .
Analogously, we introduce the unbounded operator in with the differential expression and the domain
| (2.3) |
By virtue of the Sobolev embedding theorem and the classical Green formula
| (2.4) |
the operator is closed and symmetric. The following lemma, in particular, shows that the defect index of is .
Lemma 1
The adjoint operator for has the differential expression and the domain
| (2.5) | ||||
where are cut-off functions such that
Proof. By definition, a function belongs to if and only if the following integral identity holds:
| (2.6) |
At the first step we take Based on the Green formula (2.4), we recall the Neumann boundary condition in (2.3) and apply classical results [15, §3-6, Ch. 2] on lifting smoothness of solutions to elliptic problems. In this way we conclude that and
| (2.7) |
The next step requires for results [8] of the theory of elliptic problem in domains with conical points. Indeed, regarding as the top of the ”complete cone” , that is, the punctured plane, we introduce the Kondratiev space with and as the completion of with respect to the weighted norm
| (2.8) |
where and stands for a collection of all order derivatives of . Clearly, and for .
Since , the theorem on asymptotics [8], see, e.g., [25, §3.5, §4.2, §6.4] and also the introductory chapters in the books [25, 10], gives the representation
| (2.9) |
as well as the inclusion with any and the estimate
| (2.10) |
Hence, the sum
| (2.11) |
belongs to and still solves the problem (2.7) with a new right-hand side in the Poisson equation. Thus, referring to [15, §9, Ch. 2] we have and, therefore, falls into the linear set (2.5).
2.2 The generalized Green formula
The norm, cf. the left-hand side of (2.10),
| (2.12) |
brings Hilbert structure to the linear space (2.5) denoted by . By , we understand the direct product
| (2.13) |
where is the linear space (2.2) with the Sobolev -norm. Moreover, taking into account the right-hand sides of the equations (1.20) and (1.21) we supply the vector Lebesgue space with the special norm
| (2.14) |
with
We also introduce two continuous projections by the formulas
| (2.15) | ||||
where and , are attributes of the decomposition (2.9) of .
The next assertion is but a concretization of a general result in [27, 24, 26], see also [25, §6.2], however we give a condensed and much simplified proof for reader’s convenience.
Proposition 3
For and in , the generalized Green formula
| (2.16) | ||||
is valid, where stands for the natural scalar product in .
2.3 Self-adjoint extensions
Calculations in Section 2.2 detect the defect index of the operator with the differential expression in the Hilbert space , see (2.14). Hence, this operator admits a self-adjoint extension , that is, and .
Since is a restriction of , we conclude that
| (2.19) |
where a linear subspace of dimension must be chosen such that in accord with Proposition 3
| (2.20) |
The symplectic form (2.16) is actually defined on the factor space because the generalized Green formula demonstrates that
| (2.21) |
Description of null spaces of a symplectic form in Euclidean space is a primary algebraic question, cf. [13], but it gives a direct identification of all self-adjoint extensions of our operator , see [1], [31], [7] and, e.g., [29, 3, 22].
Proposition 4
Let be an orthogonal decomposition of and let be a symmetric invertible operator in . The restriction of the operator onto the domain
| (2.22) |
is a self-adjoint extension of the operator in . Any self-adjoint extension of can be obtained in this way.
Remark 5
If we put
| (2.23) |
then the self-adjoint extension given in Proposition 4 is nothing but the set of the two-dimensional Neumann problem
| (2.24) |
and the one-dimensional mixed boundary-value problems
| (2.25) |
These problems are independent and are posed on the spaces and , respectively. The corresponding operator has the kernel spanned over the constant vectors .
Remark 6
In the case we obtain a self-adjoint extension which gives rise to the Neumann problem (2.24) and a set of the Dirichlet problems
In Section 3 we come across a self-adjoint extension (the superscript will appear in Section 3.2) with the following attributes in (2.22):
| (2.26) |
where is a symmetric non-degenerate matrix of size . In the next section we will give a different formulation of the abstract equation
| (2.27) |
which will help us to study the spectrum of
2.4 The differential problem with point conditions
Let and be projections defined in (2.15). Following [23, 22, 28], we rewrite relations imposed on and in (2.22) according to (2.26), as the point conditions
| (2.28) | ||||
| (2.29) |
We also will deal with the inhomogeneous equation
| (2.30) |
The problems
| (2.31) | ||||
| (2.32) |
with the point conditions (2.29), (2.30) give rise to the continuous mapping
| (2.33) |
Remark 7
Proposition 8
The operator in (2.33) is Fredholm of index zero.
Proof. The point conditions in Remark 6 generate the Fredholm operator of index zero
| (2.34) |
The operator (2.33) is a finite dimensional, i.e. compact, perturbation of (2.34) and thus keeps the Fredholm property. Since , the generalized Green formula (2.16) can be written in the symmetric form reflecting the particular point conditions (2.28), (2.29)
| (2.35) | ||||
and hence an argument in [15, Sect. 2.2.5, 2.5.3], cf. [25, §6.2], shows that
| (2.36) |
Let be the generalized Green function of the Neumann problem (2.24), see, e.g., [32], namely a distributional solution of
| (2.37) | |||
where is the Dirac mass. We put and write
| (2.38) |
The -matrix with entries is symmetric, see [4, Sect. 2.2]. We compose the vectors
| (2.39) |
which fall into the subspace see (2.33), because
| (2.40) | ||||
Here, is the natural basis in .
Let be a subspace spanned over the vector and with the orthogonal projector onto . We also introduce the diagonal matrix
| (2.41) |
Theorem 9
Proof. We search for a solution of the problem in the form
| (2.43) |
where is given in (2.39) and with
| (2.44) |
In view of (2.37)-(2.39) these functions must satisfy the problems
| (2.45) | ||||
Under the condition
| (2.46) |
the problems (2.45) have unique solutions (2.44) but with arbitrary constant . The vector function (2.43) fulfils the point condition (2.29) while (2.30) turns into
| (2.47) |
Applying the projector , we annul the term in (2.47) and determine , thanks to our assumption on the mapping (2.42). Then the equation (2.46) gives the remaining part of the coefficient vector in (2.43). Recalling (2.47) yields a value of .
2.5 The variational formulation of the problem with point conditions
Similarly to [23, 22, 28] we associate the problem (2.29)-(2.32) with the quadratic form
| (2.48) | ||||
defined properly in the subspace of the Hilbert space (2.13). We call (2.48) an energy functional for the problem with point conditions.
Remark 10
Theorem 11
Proof. Calculating the variation of the functional (2.48), we obtain
We make use of the generalized Green formula (2.16) while interchanging positions of and . Recalling the relation and the point condition (2.29) for , we have
| (2.49) | ||||
Here we, in particular, used the relation (2.29). It also should be mentioned that all functions are real as well as the matrix and the vector .
We see that a solution of the problem (2.29)-(2.32), annuls the variation (2.49) of the functional (2.48). On the other hand, for any test vector , the expression with the stationary point of vanishes; in particular, taking brings the differential equations in (2.31) and (2.32). Thus, the last scalar product in (2.49) is null and (2.30) is fulfilled because It remains to mention that the boundary conditions (1.22), (1.23) and the point condition (2.29) are kept in the space .
3 Determination of parameters of an appropriate hybrid model
3.1 The boundary layer phenomenon
An asymptotic analysis performed in [4] gave a detailed description of the behavior of solutions to the stationary problem in near the junction zones. The internal constitution of the boundary layers which appear in the vicinity of the sockets and are written in the rapid variables
| (3.1) |
depends crucially on the exponent in (1.13). In the case , see (1.14), the transmission conditions (1.11), (1.12) decouple and the coordinate dilation leads to two independent limit problems in the semi-infinite cylinder and the perforated layer . In [4, Sect. 2.4], we have examined these problems, namely the Neumann problem
| (3.2) | ||||
where is the outward normal derivative, and the mixed boundary-value problem
| (3.3) | ||||
We now point out several special solutions of these problems, that we need in the sequel. First of all, the homogeneous () problem (3.2) has a constant solution, say , and the problem (3.3) with is also satisfied by . The homogeneous () problem (3.3) admits a solution with the logarithmic growth at infinity
| (3.4) |
where is the logarithmic capacity of the set . Note that the function in (3.4) is independent of and is called the logarithmic capacity potential, see [30, 12]. Finally, according to [4, Lemma 6],
| (3.5) |
and a solution of the Neumann problem (3.2) with the datum in (3.5) can be found in the form
| (3.6) |
3.2 Individual choice of the self-adjoint extension
Applying the method of matched asymptotic expansions, see for example [33, 11], [16, Ch. 2], we take some functions and write the outer expansions in the plate and the rod near but outside the socket
| (3.7) | ||||
| (3.8) |
Here, ellipses stand for higher order terms of no importance in our asymptotic procedure. The inner expansions in the immediate vicinity of the sockets are composed from the above described solutions of the problems (3.3) and (3.2)
| (3.9) | ||||
| (3.10) |
We emphasize that, by definition of those solutions, the transmission condition (1.12) is wholly satisfied by (3.9) and (3.10) but the condition (1.12) with the reasonable precision
Comparing (3.7) with (3.9) and (3.8) with (3.10) yields the equations
| (3.11) | ||||
Excluding the coefficients and , we derive the relations
| (3.12) | ||||
We also introduce the diagonal -matrix
| (3.13) |
which is positive definite for with a small and further substitutes for in (2.26). Here, is the unit -matrix and .
3.3 The spectral problem
Based on results in Section 2, we give two models of the spectral problem (1.6)-(1.12). First, we use the introduced self-adjoint operator and write the equation
| (3.14) |
Second, we supply the problems (1.20), (1.22) and (1.21), (1.23) with the point conditions (2.28), (2.29) and formulate the abstract equation
| (3.15) |
where the operator involves the point condition (2.28) with the operator in (3.13) while zero at the last position in (3.15) indicates that this condition is homogeneous, namely in (2.30).
According to Remark 7, the spectral equations (3.14) and (3.15) are equivalent with each other. Unfortunately, the operator intended to model the problem (1.6)-(1.12) with the positive spectrum (1.18), is not positive definite in view of Remark 10 and formula (2.48) involving the negative definite matrix . However, in Remark 5 we mentioned the positive operator with the attributes (2.23) in (2.22) which differs from the operator only in subspace of dimension . Thus, the max-min principle, cf. [2, Thm. 10.2.2] demonstrates that the total multiplicity of the negative part of the spectrum of cannot exceed . In the next section we will construct asymptotics of negative eigenvalues which we call parasitic.
3.4 Asymptotics of parasitic eigenvalues
We will need the fundamental solution of the operator in the plane . Its expression is well known, in particular
| (3.16) |
but exact values of and are of no importance here. Let
| (3.17) |
We set
| (3.18) | ||||
| (3.19) |
where the column and the number are to be determined and for and for . Clearly, the functions (3.18), (3.19) satisfy the boundary conditions (1.22), (1.23) and leave small discrepancies in the differential equations (1.20), (1.21) with the spectral parameter . It should be mentioned that the exponential decay of the boundary layer terms in (3.18) and (3.19) is due to the negative value of in (3.17).
The vector has the following projections:
| (3.20) | ||||
where Hence, the point condition (2.29) is fulfilled while, in view of (3.13) and (3.20), the condition (2.28) converts into
or, what is the same,
| (3.21) |
Since both matrices are diagonal, the system (3.21) splits into independent transcendental equations. The small factor of the exponent allows us to apply the implicit function theorem and to obtain the solutions
| (3.22) |
We have derived ”good approximations” for negative eigenvalues of the spectral equations (3.14) and (3.15). Recalling that their number cannot exceed we come in position to formulate an assertion on the whole negative part of the spectrum.
Proposition 12
We will outline the proof in Remark 17. Notice that the negative eigenvalues are situated very far away from the positive part of the spectrum, that is, outside the scope of the asymptotic models under consideration.
4 Justification of the asymptotic models
4.1 The first convergence theorem
Let be an eigenvalue of the self-adjoint operator in (3.14). The corresponding eigenvector can be normed as follows:
| (4.1) |
Assuming that
| (4.2) |
in particular, rejecting negative eigenvalues in Proposition 12, we recall the Kondratiev theory used in Section 2.1. Then, a solution of the problem (1.20), (1.22) admits the decomposition (2.9) with the ingredients and while
| (4.3) |
where is given in (2.11).
Furthermore, a solution of the ordinary differential equation (1.21) with the Dirichlet condition (1.23) falls into the Sobolev space and fulfills the estimate
| (4.4) |
The inequalities (4.2)-(4.4) help to conclude the following convergence along an infinitesimal positive sequence :
| (4.5) | ||||
This and the embeddings imply the convergence of the projections (2.15)
| (4.6) |
We emphasize that formulas in (4.5) guarantee the strong convergences in and in so that the normalization condition (4.1) is kept by the limit . Moreover, the differential equations (1.20), (1.21) with and the boundary conditions (1.22), (1.23) for are passed to the limits
| (4.7) |
In order to formulate the next assertion it suffices to mention that the point conditions (2.28), (2.29) with the matrix (3.13) containing the big component turn in the limit into the relations
| (4.8) |
These provide the self-adjoint extension with the attributes (2.23) in (2.22) that corresponds to the Neumann and mixed boundary-value problems (2.24) and (2.25). The spectra and of the above mentioned problems are united into the common monotone sequence
| (4.9) |
Here, eigenvalues are listed while counting their multiplicity in.
Theorem 13
If an eigenvalue of the operator , cf. (3.14) and (3.15), and the corresponding eigenvector fulfil the requirements (4.2) and (4.1), then the limits and in (4.5) along an infinitesimal sequence are an eigenvalue of the operator described in Remark 5 and the corresponding eigenvector normed in the space , see (2.14).
4.2 The second convergence theorem
In the next section we will verify that entries of the eigenvalue sequence (1.18) of the original problem (1.6)-(1.12) in the junction satisfy the inequalities
| (4.10) |
with some positive and which depend on the eigenvalue number but are independent of . The corresponding eigenfunction is subject to the normalization condition (1.19). We introduce the functions
| (4.11) | ||||
| (4.12) |
and write
| (4.13) | ||||
Here, we used formulas (1.17) and (1.13), (1.14) while taking the relation on into account. Hence, the vector function satisfies the estimate
| (4.14) |
A similar calculation gives us the formula
Moreover, the Poincaré inequalities in and show that the functions
which are of mean zero in and , respectively, enjoy the relations
| (4.15) | ||||
Hence, we obtain
| (4.16) | ||||
To estimate the norm , we apply the weighted inequality in [4, Thm. 9]
| (4.17) | |||
where and is independent of and . Since in the socket , we recall (1.16), (4.10) and conclude that
| (4.18) | ||||
The estimates (4.10) and (4.13), (4.14) provide the following convergence along an infinitesimal sequence :
| (4.19) | ||||
We compose a test function from components and . Then according to (1.17) and (4.11), (4.12), we transform the integral identity into the formula
We multiply this with and perform the limit passage along the sequence to obtain
| (4.20) |
Thanks to [15, §9 Ch. 2], the weak solution of the variational problem (4.20) where , falls into and satisfies the Neumann problem (1.20), (1.22) with . We emphasize that is dense in so that any test function is available. At the same time, vanishes near the points and . Hence, we may conclude that and the differential equation (1.21) with . However, the boundary conditions
| (4.21) |
still must be derived. The Dirichlet condition in (4.21) is inherited from the conditions and . To conclude with the Neumann condition, we observe that the inequality (4.18) allows us to repeat the above transformations with the ”very special” test vector function
| (4.22) |
where and are taken from (2.5) and (3.19). As a result, the obtained information on and reduces the integral identity (4.21) with (4.22) to the formula
We are in position to formulate the convergence theorem.
4.3 An abstract formulation of the original problem
In the Hilbert space we introduce the scalar product
| (4.23) |
and positive, symmetric and continuous, therefore, self-adjoint operator ,
| (4.24) |
Bilinear form on the right-hand side of (4.23) are defined in (1.17). Comparing (1.16) with (4.23), (4.24), we see that the variational formulation of the problem (1.6)-(1.12) in is equivalent to the abstract equation
| (4.25) |
with the new spectral parameter
| (4.26) |
The operator is compact and, hence, the essential spectrum of consists of the only point , see, e.g., [2, Thm. 10.1.5], while the discrete spectrum composes the positive monotone infinitesimal sequence
| (4.27) |
The following assertion is known as the lemma on ”near eigenvalues and eigenvectors” [34] following directly from the spectral decomposition of resolvent, see, e.g., [2, Ch. 6].
Lemma 15
Let and satisfy
| (4.28) |
Then there exists an eigenvalue of the operator such that
| (4.29) |
Moreover, for any , one finds coefficients to fulfil the relations
| (4.30) |
where are all eigenvalues in the segment and are the corresponding eigenvectors subject to the normalization and orthogonality conditions
| (4.31) |
4.4 Detecting eigenvalues with prescribed asymptotic form
Let and be an eigenvalue of the equation (3.15) and the corresponding eigenvector enjoing the normalization and orthogonality conditions
| (4.32) |
where denotes the scalar product in the Lebesgue space induced by the norm (2.14). In Lemma 15 we set
| (4.33) |
and build an asymptotic approximation of an eigenfunction of the problem (1.6)-(1.12) in the junction (1.4). To mimic the method of matched asymptotic expansions, we use asymptotic structures with ”overlapping” cut-off functions, see [16, Ch.2], [22, 18] etc., namely in addition to the functions in (2.5) and in (3.19) we introduce
| (4.34) | ||||
Radius is chosen such that on
The functions and are determined by formulas
| (4.35) | ||||
| (4.36) | ||||
| (4.37) |
where ingredients are taken from (3.9), (3.10) and is a function with compact support such that
| (4.38) |
The latter condition and the cut-off functions in (4.36), (4.37) assure that and coincide with each other on , cf. (1.11), and the composite function falls into
First of all, we compute the scalar products . To this end, we observe that, according to (3.12), we have
| (4.39) |
The estimate (1.9) applied in the problem (1.20), (1.21) shows that
| (4.40) |
Moreover, a solution of (1.21), (1.23) satisfies
| (4.41) |
while the last bound is due to the estimate (4.40) and the small factor on the right of (4.39).
Recalling that and , see (3.11), we rewrite (4.36) as follows:
We list the estimates
| (4.42) | ||||
These must be commented. Notice that the factor comes due to integration in . In the first estimate we took into account that outside the disk where and applied the weighted inequality (1.9) for with . The second and fourth estimates were derived by a direct calculation of norms and using the decomposition (3.4) of and the bound for , cf. (4.39)-(4.41). The third estimate is obtained by the coordinate change , see (3.1).
The above listed estimates support the following relation:
| (4.43) | |||
Moreover,
| (4.44) | ||||
| (4.45) |
Indeed, in (4.44) we got rid of by using the second estimate (4.42) and evaluate by means of the weighted inequality (1.9) again. To conclude (4.45), we observed additionally that
because is a solution of the problem
with a right-hand side which is caused by abolition of and therefore has the -norm of order .
In a similar way but with much simpler calculations (recall that is a smooth function on ) we derive the inequalities
| (4.47) |
Notice that the factor is compensated due to the relation . According to (1.17), (4.23) and (4.32), (2.14), the inequalities (4.46) and (4.47) lead us to
| (4.48) |
Now we evaluate the norm in (4.28), namely
| (4.49) |
The first factor on the right is bounded, see Theorem 14, and the second one does not exceed owing to (4.48) with . Using a definition of a Hilbert norm together with formulas (4.23) and (4.24), we obtain that the last factor is equal to
| (4.50) |
where supremum is computed over the unit ball in . Inequality (4.17), see [4, Thm. 9], indicates bounds for weighted Lebesgue norms of .
We insert representations (4.36) and (4.37) into the last expressions between the modulo sign in (4.50) and detach the elementary term
| (4.51) |
We derived the estimate in the same way as above, that is, the information on and the coordinate change .
Other ingredients are sufficiently smooth while integration by parts and commuting the Laplace operator with cut-off functions yield
| (4.52) | ||||
| (4.53) | ||||
| (4.54) | ||||
| (4.55) | ||||
Note that integrals over the surfaces and vanish due to our choice of cut-off functions and boundary conditions for and , see Section 1.2 and 3.1, respectively. We will estimate all scalar products in (4.52)-(4.54) and explain how their sum converts into .
In view of the equation (1.20) the first term in (4.52) is null. The next term in (4.52) is obtained from the first and third terms in (4.36) after commuting the Laplace operator with the cut-off function ; notice that
| (4.56) | ||||
Since , see (2.9), and supports of coefficients in the differential operator belong to the disk , see (4.34), the direct consequence of the one-dimensional Hardy inequality
| (4.57) | ||||
provides that
| (4.58) | |||
Here, we took into account that for . The first (long) multiplier in the middle of (4.58) is written according to the relation , formula for in (4.56) and weights in (4.57). The factor is due to integration in and finally the weights and from (4.17) were considered. It should be also mentioned that and, therefore, for .
Dealing with (4.53) we observe that the Laplace equation for in (3.3) annuls the first term . Supports of coefficients of , see (4.56), are located in the annulus . Hence, using the decay rate in of the remainder , see (3.4), we derive the following estimates for the second and third terms in (4.53):
| (4.59) | ||||
Here, we applied (4.17) again and recalled that . It should be mentioned that and , respectively, involve the commutator and the multiplication operator acting on and the last term in (4.36). In this way the sum exhibits the whole part of generated by (4.36).
Referring to (4.54), we see that in view of (1.21). The second term in (4.54) meets the estimate
Here, came from (4.54), the relations and were used, the factor is due to integration over supp and finally the direct consequence of the Newton-Leibnitz formula
| (4.60) |
together with the inequality (4.17) were applied.
Similarly to the above considerations, the first term in (4.55) vanishes, cf. (3.2), and the other couple of terms can be estimated as follows:
Here, we took into account the exponential decay in (3.6) together with formulas (4.60) and (4.17).
In the same way as above we detect a proper redistribution of commutators of with and conclude that equals a part of generated by (4.37).
4.5 Theorem on asymptotics
We are in position to conclude with the main assertion of the paper.
Theorem 16
Proof. The result directly stems from (4.61) and all the previous considerations. We still need to utter several important remarks, in order to complete the proof. First, if is eigenvalue of multiplicity , then Lemma 15 provides us with different eigenvalues satisfying (4.61) with a bigger constant . Indeed, owing to (4.48), the constructed approximate eigenfunctions are almost orthonormalized, that is
Moreover, setting in (4.30), we see that their projection onto the linear hull are linear independent for a small and a big that is possible in the case only. Thus, changing in (4.61) gives us at least desired eigenvalues.
Second, since each eigenvalue in (4.62) has an eigenvalue in its small neighborhood and for , we obtain and confirm the assumption (4.10) so that Theorem 14 becomes true.
Finally, we assume without loss of generality that is fixed such that the eigenvalues and of the operator obey the relation . If it happens that the index of the eigenvalue in the vicinity of is strictly bigger that , then, for an infinitesimal positive sequence , we have with some . We can apply Theorem 14 and conclude that as is an eigenvalue in the interval , while the limits of (4.11), (4.12) constructed from the corresponding eigenfunctions are orthogonal in to the vector eigenfunctions , of the operator . Since belongs to the spectrum of but , the latter is absurd.
Remark 17
5 Final remarks
5.1 Simplified and rough asymptotics
Since the point condition (2.28) contains the big parameter in the matrix (3.13), it is straightforward to write an asymptotics in for eigenvalues (1.18). In view of the precision estimate (4.63) in Theorem 16 it suffices to find a decomposition of eigenvalues in the model problem (1.20)-(1.23), (2.28), (2.29), in particular, of the projections (2.15) of the corresponding eigenvectors.
Let us demonstrate the simplest ansatz for the first eigenvalue
| (5.1) |
The corresponding eigenvector of the model problem is searched in the asymptotic form
| (5.2) |
while the absence in (5.1) of the term clearly requires that
| (5.3) |
and therefore
| (5.4) |
Hence, a problem to determine involves the Neumann problem
| (5.5) |
which is derived by inserting (5.1)-(5.3) into the equations (1.20)-(1.22) and extracting terms of order . Furthermore, we obtain from the point condition (2.28) with the matrix from (3.13) that
| (5.6) |
The generalized Green formula (2.16) delivers the compatibility condition in this problem, namely
Note that we obtained in the ansatz (5.1) the first term
| (5.7) |
which does not contain much information about the rod elements of the junction , namely only their number .
In order to construct the second term , we, first of all, observe that a solution of the problem (5.5) can be subject to the orthogonality condition
| (5.8) |
and, thus, formulas (5.5), (5.6) and (2.37), (2.39) lead to the representation
so that
| (5.9) |
Owing to (5.8), the compatibility condition in the problem
reads
and implies
| (5.10) |
The diagonal matrices and depend on length and the reduced cross-section of the rod , while the matrix reflects disposition of the junction points in the base of the plate .
5.2 The homogeneous junction
By setting in (1.13) the restrictions (1.15), the problem (1.6)-(1.12) reduces to the mixed boundary-value problem
| (5.12) | ||||
stated in the intact domain , cf. (1.4) and (1.1), (1.3). An asymptotic analysis of the stationary problem (5.12) with a source term instead of has been developed in [4, Sect. 2]. Let us outline certain peculiarities of asymptotic models in the case (1.15), which are to be derived along the same scheme as in Sections 3 and 4.
As was mentioned in Section 1.2, a specific feature of the homogeneous junction is that the limit operator decouples, see (5.17), i.e., instead of the connected skeleton in fig. 2,a, we obtain in the limit the disjoint domain and intervals as drawn in fig. 2,b.
Both the asymptotic models can be applied for the stationary problem of type (1.6)-(1.12) with the source term instead of on the right-hand side of the Poisson equations. However, a result in [4] displays an explicit rational dependence on of the corresponding solution that furnishes its complete asymptotic form in finite steps. Moreover, it reduces an evident importance of the models for the spectral problem (1.6)-(1.12) where an argument based on the big parameter in (3.13) and (2.28) detects the holomorphic dependence on which clearly cannot be described in finite steps.
Trying to formulate point conditions connecting the limit problems (1.20), (1.22) and (1.21), (1.23) in the skeleton of the junction , see fig. 1 and 2, we need a special solution of the homogeneous Neumann problem for the Laplace equation in the union of a layer and a semi-infinite cylinder. As was shown in [4], this solution gets the asymptotic behavior
| (5.13) | ||||
| (5.14) |
It should be noticed that a proper choice of the coordinate origin eliminates the constants because the change with provides the formulas
In what follows we assume that the points and, therefore, the coordinates are fixed such that in (5.13).
Repeating the matching procedure performed in Section 3.2, we keep the expansions (3.7), (3.8) but, according to (5.13), (5.14), replace (3.9), (3.10) with the following ones:
Thus, we obtain the relations
| (5.15) |
which look quite similar to (3.12) but we have the small factor on the derivatives , that is, on the projection . To perform the correct limit passage , we recall our previous asymptotic analysis in [4, Sect. 2] and make the substitution
| (5.16) |
As a result, we conclude the point conditions
| (5.17) |
corresponding to the self-adjoint extension described in Remark 6. In other words, the limit spectrum (4.9) is composed from the spectrum of the Neumann problem in and the spectra of the Dirichlet problems in . It is worth to mention that, as was observed in [4], the substitution (5.16) has a clear physical reason, namely the energy functional for the problem (5.12) gets an appropriate approximation by the sum of the energy functionals for the above mentioned limit problems multiplied with the common factor .
Let us construct the first correction term in the asymptotics
| (5.18) |
of the first eigenvalue of the problem (5.12). Recalling an asymptotic procedure in [4, Section 2], we search for the corresponding eigenfunction in the form
| (5.19) | ||||
Regarding (5.19) as outer expansions, we write the inner expansions in the vicinity of the sockets as follows:
Finally, we take the representations (5.13), (5.14) and apply the matching procedure, cf. Section 3.2, to close the problem (5.5) with the asymptotic conditions near the points
| (5.20) |
Similarly to Section 5.1 the compatibility condition in the problem (5.5), (5.20) converts into the formula
Combining approaches in Section 4 and [4, Sect. 2], the asymptotic formula (5.18) can be justified by means of the estimate for the remainder.
Let us consider the problems (1.20), (1.22) and (1.21), (1.23) connected through the point conditions
| (5.21) |
where because the junction is homogeneous and the -matrix is given by formula (3.13) with instead of , see (5.13) and (3.4).
The conditions (5.21) follow immediately from (5.15) after substitution (5.16). They are involved into the symmetric generalized Green formula of type (2.35)
and therefore, all requirements in Section 2 and 3 with slight modifications apply to the model in the case (1.15), too. Moreover, a simple analysis requiring only for algebraic operations as in Section 5.1, provides the representation
| (5.22) |
of an eigenvalue in the model (1.20)-(1.23), (5.21) which is supplied with an operator of type (2.33) on the function space (2.13) with detached asymptotics.
Acknowledgements.
The work of S.A.N. was supported by grant 15-01-02175 of Russian Foundation for Basic Research. G.C. is a member of GNAMPA of INDAM.
References
- [1] Beresin F.A., Faddeev L.D., A remark on Schrödinger equation with a singular potential, Sov. Math. Doklady, 137 (5) (1961) 1011-1014.
- [2] Birman M.Sh., Solomjak M.Z., Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987.
- [3] Brüning J., Geyler V., Pankrashkin K., Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Rev. Math. Phys. 20 (1) (2008) 1-70.
- [4] Bunoiu R., Cardone G., Nazarov S.A., Scalar boundary value problems on junctions of thin rods and plates. I. Asymptotic analysis and error estimates, ESAIM: Math. Modell. Num. Anal. 48 (2014) 1495-1528.
- [5] G. Buttazzo, G.Cardone, S.A.Nazarov, Thin Elastic Plates Supported over Small Areas. I: Korn’s Inequalities and Boundary Layers, J. Convex Analysis 23 (1) (2016), 347-386.
- [6] G. Buttazzo, G.Cardone, S.A.Nazarov, Thin Elastic Plates Supported over Small Areas. II: Variational-asymptotic models, J. Convex Analysis 24 (3) (2017) 819-855.
- [7] Colin De Verdière Y., Pseudo-Laplaciens II, Ann. Inst. Fourier 33 (2) (1983) 87-113.
- [8] Kondratiev V.A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967) 227-313.
- [9] Kozlov V., Maz’ya V., Movchan A., Asymptotic analysis of fields in multi-structures. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999.
- [10] Kozlov V.A., Maz’ya V.G., Rossmann J., Elliptic boundary value problems in domains with point singularities. Providence: Amer. Math. Soc., 1997.
- [11] Il’in A.M., Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, 102. American Mathematical Society, Providence, RI, 1992.
- [12] Landkof N.S., Foundations of modern potential theory. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972.
- [13] Lang S., Algebra, Graduate text in Mathematics 211, Springer 2002.
- [14] Lions J.-L., Some more remarks on boundary value problems and junctions. Asymptotic methods for elastic structures (Lisbon, 1993), 103–118, de Gruyter, Berlin, 1995.
- [15] Lions J.L., Magenes E., Non-homogeneous boundary value problems and applications, Springer-Verlag, New York-Heidelberg, 1972.
- [16] Maz’ya V.G., Nazarov S.A., Plamenevskij B.A., Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Operator Theory: Advances and Applications, 112, Birkhäuser Verlag, Basel (2000).
- [17] Nazarov S.A., Junctions of singularly degenerating domains with different limit dimensions. 2, Trudy seminar. Petrovskii. 20 (1997) 155-195 (English transl.: J. Math. Sci., 97 (3) (1999) 155–195.)
- [18] Nazarov S.A., Estimates for the accuracy of modeling boundary value problems on the junction of domains with different limit dimensions, Izv. Math., 68 (6) (2004) 1179–1215.
- [19] Nazarov S.A. Asymptotic behavior of the solution and the modeling of the Dirichlet problem in an angular domain with rapidly oscillating boundary, St. Petersburg Math. J., 19 (2) (2007) 297-326.
- [20] Nazarov S.A., Modeling of a singularly perturbed spectral problem by means of self-adjoint extensions of the operators of the limit problems, Funkt. Anal. i Prilozhen. 49 (1) (2015) 31-48 (English transl.: Funct. Anal. Appl. 49 (1) (2015) 25–39).
- [21] Nazarov S.A., Elliptic boundary value problems on hybrid domains, Funkt. Anal. i Prilozhen, 38 (4) (2004), 55-72 (English transl.: Funct. Anal. Appl. 38 (4) (2004) 283-297).
- [22] Nazarov S.A., Asymptotic conditions at a point, self-adjoint extensions of operators and the method of matched asymptotic expansions, Trans. Am. Math. Soc. 193 (1999) 77-126.
- [23] Nazarov S.A. Asymptotic solution to a problem with small obstacles, Differential equations. 31 (6) (1995) 965-974.
- [24] Nazarov S.A., Plamenevskii B.A. Selfadjoint elliptic problems with radiation conditions on the edges of the boundary, St. Petersburg Math. J. 4 (3) (1993) 569-594.
- [25] Nazarov, S.A., Plamenevsky, B.A. Elliptic problems in domains with piecewise smooth boundaries. de Gruyter Expositions in Mathematics, 13. Walter de Gruyter & Co., Berlin, 1994.
- [26] Nazarov S.A., Plamenevskii B.A. Elliptic problems with radiation conditions on edges of the boundary, Sb. Math. 77 (1) (1994) 149-176.
- [27] Nazarov S.A., Plamenevskii B.A. A generalized Green’s formula for elliptic problems in domains with edges, J. Math. Sci. 73 (6) (1995) 674-700.
- [28] Nazarov S.A., Specovius-Neugebauer M., Modeling of cracks with nonlinear effects at the tip zones and the generalized energy criterion, Arch. Rational Mech. Anal. 202 (2011) 1019-1057.
- [29] Pavlov B.S., The theory of extensions, and explicit solvable models, Russian Mathematical Surveys 42 (6) (1987) 127-168.
- [30] Pólya G., Szegö G., Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, n. 27, Princeton University Press, Princeton, N. J., 1951.
- [31] Rofe-Beketov F.S., Self-adjoint extensions of differential operators in the space of vector functions, Dokl. Akad. Nauk SSSR 184 (1969) 1034-1037; English transl.: Soviet Math.Dokl. 10 (1969) 188-192.
- [32] Smirnov V.I., A course of higher mathematics. Vol. II. Advanced calculus. Translation edited by I. N. Sneddon Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London 1964.
- [33] Van Dyke M., Perturbation methods in fluid mechanics, Applied Mathematics and Mechanics, Vol. 8 Academic Press, New York-London 1964.
- [34] Visik M. I., Ljusternik L.A., Regular degeneration and boundary layer of linear differential equations with small parameter, Amer. Math.Soc. Transl. 20 (2) (1962) 239-364.